The Structure of Groups with a Quasiconvex Hierarchy

Name: The Structure of Groups with a Quasiconvex Hierarchy | (AMS-209)
Brand: Princeton University Press
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(AMS-209)

Daniel T. Wise(Autor*in)

Princeton University Press

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Erschienen am 4. Mai 2021

376 Seiten

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Inhalt

Cover
Contents
Acknowledgments
1. Introduction
2. CAT(0) Cube Complexes
2.a Basic Definitions
2.b Right-Angled Artin Groups
2.c Hyperplanes in CAT(0) Cube Complexes
2.d Geodesics and the Metric
2.e Properties of Minimal Area Cubical Disk Diagrams
2.f Convexity
2.g Hyperplanes and Their Carriers
2.h Splaying and Rectangles
2.i Annuli
2.j Annular Diagrams and Malnormality
2.k Convex Cores
2.l Superconvexity
3. Cubical Small-Cancellation Theory
3.a Introduction
3.b Cubical Presentations
3.c Pieces
3.d Some Small-Cancellation Conditions to Bear in Mind
3.e Disk Diagrams and Reduced Disk Diagrams
3.f Rectified Disk Diagrams
3.g Gauss-Bonnet Theorem
3.h Assigning the Angles
3.i Nonpositive Curvature of Shards
3.j Tables of Small Shards
3.k Nonpositive Curvature of Cone-Cells via Small-Cancellation
3.l Internal Cone-Cells That Do Not Self-Collide
3.m More General Small-Cancellation Conditions and Involved Justification
3.n Informal Discussion of the Limits of the Theory
3.o Nonpositively Curved Angling Rules
3.p Positive Curvature along Boundary
3.q Ladder Theorem
3.r Trichotomy for Reduced Diagrams
3.s Examples
3.t Examples Arising from Special Cube Complexes
3.u Graded Small-Cancellation
3.v Some Graded Examples
3.w Graded Metric Small-Cancellation
3.x Missing Shells and Injectivity
3.y Short Innerpaths and Quasiconvexity
4. Torsion and Hyperbolicity
4.a Cones Embed
4.b Torsion
4.c Hyperbolicity
5. New Walls and the B(6) Condition
5.a Introduction
5.b Total Defects of Paths in Cones
5.c Generalization of the B(6) Condition
5.d Cyclic Quotients and the B(6) Condition
5.e Embedding Properties of the Cones and Hyperplane Carriers
5.f Defining Immersed Walls in X*
5.g No Inversions
5.h Carriers and Quasiconvexity
5.i Bigons
5.j Square Cones
5.k 1-Dimensional Linear Separation
5.l Obtaining Proper Actions on the Dual
5.m Codimension-1 Subgroup Preserved
5.n Elliptic Annuli
5.o Annular Diagrams and the B(8) Condition
5.p Doubly Collared Annular Diagrams
5.q Malnormality of Wall Stabilizers
5.r Artin Groups
6. Special Cube Complexes
6.a Immersed Hyperplanes
6.b Hyperplane Definition of Special Cube Complex
6.c Right-Angled Artin Group Characterization
6.d Canonical Completion and Retraction
6.e Double Cosets and Virtual Specialness
6.f Extensions of Quasiconvex Codimension-1 Subgroups
6.g The Malnormal Combination Theorem
7. Cubulations
7.a Wallspaces
7.b Sageev's Construction
7.c Finiteness Properties of the Dual Cube Complex
7.d Virtually Cubulated
7.e Sparse Complexes
7.f Useful Subwallspaces
7.g Cubulating Amalgams
8. Malnormality and Fiber-Products
8.a Height and Virtual Almost Malnormality
8.b Fiber-Products
8.c Graded Systems
9. Splicing Walls
9.a Finite Cover That Is a Wallspace
9.b Preservation of Small-Cancellation and Obtaining Wall Convexity
9.c Obtaining the Separation Properties for Pseudographs
10. Cutting X*
10.a Hierarchies of Cubical Presentations
10.b Inflations
10.c Some Persistent Properties
10.d Additional Splitting along Conepoints
11. Hierarchies
12. Virtually Special Quotient Theorem
12.a Malnormal Special Quotient Theorem
12.b Proof of the Special Quotient Theorem
12.c Adding Higher Grade Relators
12.d Controlling Intersections in Quotient
13. Amalgams of Virtually Special Groups
13.a Virtually Special Amalgams
14. Large Fillings Are Hyperbolic and Preservation of Quasiconvexity
14.a Hyperbolic Fillings
14.b Quasiconvex Image
15. Relatively Hyperbolic Case
15.a Introduction
15.b Parabolic Fillings That Are Virtually Special
15.c Separability for Relatively Hyperbolic Hierarchies
15.d Residually Verifying the Double Coset Criterion
15.e Relative Malnormality and Separability
15.f The Hierarchy in the Relatively Hyperbolic Setting
16. Largeness and Omnipotence
16.a Virtual Separation and Largeness
16.b Omnipotence
17. Hyperbolic 3-Manifolds with a Geometrically Finite Incompressible Surface
17.a Some Background on 3-Manifolds
17.b Aparabolic Hierarchy
17.c Virtual Specialness of Hyperbolic 3-Manifolds with Boundary
17.d Cutting All Tori with First Surface
18. Limit Groups and Abelian Hierarchies
18.a Limit Groups
18.b Abelian Hierarchies
19. Application Towards One-Relator Groups
19.a Overview
19.b The Magnus-Moldavanskii Hierarchy
19.c Quasiconvexity Using the Strengthened Spelling Theorem
19.d Staggered 2-Complex with Torsion
20. Problems
References
Index

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